Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?

Edit: Alexander Campbell points out in the comments that a reference for this, in the case ${\cal V}=\rm Set$, is Claudio Pisani's paper Sequential multicategories. Following is a sketch of the argument.

Consider the category $\mathcal{V}\text{-}\mathrm{Mult}$ of $\mathcal{V}$-enriched multicategories. This includes the category $\mathcal{V}\text{-}\mathrm{MonCat}_l$ of monoidal $\mathcal{V}$-categories and lax monoidal functors as a full subcategory, where the underlying multicategory of $\mathcal{C}$ has multimorphisms defined by $U\mathcal{C}(c_1,\dots,c_n; x) = \mathcal{C}(c_1\otimes\cdots\otimes c_n, x)$.

Furthermore, I claim the category $\mathcal{V}\text{-}\mathrm{Mult}$ has a "pointwise" monoidal structure, which $\mathcal{V}\text{-}\mathrm{MonCat}_l$ is closed under. Given $\mathcal{V}$-multicategories ${\cal C,D}$, define the objects of ${\cal C\otimes D}$ to be pairs $(c,d)$ of objects of $\cal C$ and $\cal D$ respectively, with

$$({\cal C\otimes D})((c_1,d_1),\dots,(c_n,d_n);(x,y)) = {\cal C}(c_1,\dots,c_n;x) \otimes {\cal D}(d_1,\dots,d_n;y).$$

Note that this is not the "Boardman-Vogt tensor product" of multicategories/operads; in particular the latter only makes sense for symmetric multicategories, but this one works even for non-symmetric ones.

Now your claim can be re-expressed as

$$\mathcal{V}\text{-}\mathrm{Mult}({\cal C}, [{\cal D,V}]) \cong \mathcal{V}\text{-}\mathrm{Mult}({\cal C\otimes D,V}),$$

i.e. that the underlying multicategory of the Day convolution monoidal category $[{\cal D,V}]$ is an internal-hom for this tensor product on $\mathcal{V}\text{-}\mathrm{Mult}$. This should be straightforward because both of these multicategories have a direct description.

Of course the objects of the Day convolution monoidal category are the functors between the underlying ordinary categories $\cal D,V$. You might expect at first that the objects of the internal-hom in ${\cal V}\text{-}\mathrm{Mult}$ should instead be multicategory functors $\cal D\to V$, but in fact this is not true, because the "unit multicategory" is not the same as the "multicategory freely generated by an object", and it is maps out of the latter that detect the objects of the internal-hom. In the multicategory $\cal O$ freely generated by an object there are no $n$-ary morphisms for $n\neq 1$, and the unique object has only its identity unary morphism; thus a multicategory functor $\cal O \otimes D\to V$ acts only on the objects and unary morphisms of $\cal D$, and hence is also just a functor between underlying ordinary categories.

Now the $n$-ary morphisms in the internal-hom are the same as morphisms $\cal T_n\otimes D \to V$, where $\cal T_n$ is the "free-living $n$-ary morphism". Thus a morphism $(F_1,\dots, F_n) \to G$ consists of, for each $n$-ary morphism $(d_1,\dots,d_n) \to y$ in $\cal D$, an $n$-ary morphism $(F_1(d_1),\dots,F_n(d_n)) \to G(y)$ in $\cal V$, subject to naturality with respect to unary morphisms (because of the identity arrows in $\cal T_n$). In the enriched case we have to rewrite that as a definition of a hom-object:

$$[{\cal D,V}](F_1,\dots,F_n;G) = \int_{d_1,\dots,d_n,y} [{\cal D}(d_1,\dots,d_n;y),{\cal V}(F_1(d_1),\dots,F_n(d_n);G(y))].$$

On the other hand, the Day convolution tensor product is defined as

$$(F_1\otimes F_2)(y) = \int^{d_1,d_2} F_1(d_1)\otimes F_2(d_2) \otimes {\cal D}(d_1\otimes d_2,y)$$

so that

$$\begin{align}[{\cal D,V}](F_1\otimes F_2,G) &= \int_{y} {\cal V}\left(\int^{d_1,d_2} F_1(d_1)\otimes F_2(d_2) \otimes {\cal D}(d_1\otimes d_2,y),G(y)\right)\\ &\cong \int_{y,d_1,d_2} {\cal V}(F_1(d_1)\otimes F_2(d_2) \otimes {\cal D}(d_1\otimes d_2,y),G(y))\\ &\cong \int_{y,d_1,d_2} [{\cal D}(d_1\otimes d_2,y),{\cal V}(F_1(d_1)\otimes F_2(d_2),G(y))] \end{align}$$

which coincides exactly with the other version when $n=2$ and $\cal D,V$ are representable. A similar argument works for arbitrary $n$. So far we haven't used that $\cal D,V$ are monoidal rather than arbitrary multicategories, but apparently to construct composition one needs $\cal D$ to be at least promonoidal.

Said slightly differently, this is the observation that monoidal profunctors $J\colon A \to B$, that is lax monoidal functors $J\colon A^{op} \times B \to V$, correspond (up to isomorphism) to lax monoidal functors $j\colon B \to \hat A$, where $\hat A = [A^{op}, V]$ is equipped with Day-convolution. Each corresponding pair $J$ and $j$ comes with an isomorphism of monoidal profunctors

$$\hat A(ya, jb) \simeq J(a, b),$$

where $y\colon A \to \hat A$ is the yoneda embedding (which is pseudomonoidal wrt. Day-convolution).

This you can think of as a "monoidal Yoneda's lemma" and, formally, as being part of some kind of yoneda structure. This is the main motivation for https://arxiv.org/abs/1511.04070.

To formalise the above you need a setting allowing two types of morphism (mon. profunctors and lax mon. functors). This is why I used a form of generalised double categories $K$, instead of 2-categories which are traditionally used for yoneda structures.

One of main results states that for a "reasonable" double monad $T$ on $K$, given a yoneda embedding $y\colon A \to \hat A$ in $K$, a $T$-algebra structure on $A$ induces such a structure on the presheaf object $\hat A$, making $y$ into a yoneda embedding in the "double category" $Alg(T)$ of $T$-algebras. Taking $T =$ "free monoidal $V$-cats"-monad then recovers the correspondence above.

You can also apply this result to get yoneda embeddings for double categories and for categories $A\colon S^{op} \to Cat$ indexed over some small category $S$. In both cases you get presheaf objects that are simpler than the classical ones (given by resp. Bob Pare (http://www.tac.mta.ca/tac/volumes/25/17/25-17.pdf) and e.g. Mark Weber (https://link.springer.com/article/10.1007/s10485-007-9079-2) )

You should also be able to get/recover yoneda embeddings for "cocomplete modular" closure/topological/approach spaces (in the sense of https://arxiv.org/abs/1704.00209) by taking $T$ to be some extension of the power-set or ultrafilter monad, but I haven't had the time to work that out.

Another main result proves that the "lifted" algebraic yoneda embedding defines $\hat A$ as the free cocompletion of $A$ in $Alg(T)$. In the monoidal case this recovers a result of Kelly and Im given in https://www.sciencedirect.com/science/article/pii/0022404986900058.

I hope this helps. It has been good for me to find references to similar results here.